Bending and twisting in electromagnetic waveguides - Michele Zaccaron (Aix-Marseille Université)

A 3D straight, non-twisted waveguide is the cartesian product R × ω, where ω, the
cross-section, is a bounded domain of R2.
In non-dispersive media enclosed by a perfect conductor, the macroscopic time-dependent
Maxwell’s equations are deeply related to the Maxwell operator
A = i ( (0, ε^−1 curl) , (−μ^−1 curl,0)).
where ε and μ are the electric permittivity and magnetic permeability of the medium,
respectively.
Supposing we have a homogeneous and isotropic material, the spectrum of the Maxwell
operator in a perfectly conducting straight waveguide with a constant twist (which can be
zero) is entirely essential, lies on the real line and is symmetrical with respect to zero,
forming a gap around the origin.
In this talk we will present new results concerning the geometrical deformations of bend-
ing and twisting of waveguides, and their consequent effects on the spectrum of the Maxwell
operator A. In particular we will provide some sufficient conditions on the curvature and
twist at infinity so that the essential spectrum is preserved. The proof will involve a Birman-
Schwinger type principle, which has an interest on its own. Finally, we will discuss the
possible presence of geometrically induced discrete eigenvalues in the gap, and in the case of
zero twist we will provide a sufficient condition on the curvature to ensure their existence.